$(1 - \cos^2\theta)(\csc^2\theta) = \; ?$
Solution: We can use the identity ${\sin^2 \theta} + {\cos^2 \theta} = 1$ to simplify this expression. $1$ ${\sin\theta}$ ${\cos\theta}$ $\theta$ We can see why this is true by using the Pythagorean Theorem. So, $1 - \cos^2\theta = \sin^2\theta$ Plugging into our expression, we get $ (1 - \cos^2\theta)(\csc^2\theta) = (\sin^2\theta)(\csc^2\theta) $ To make simplifying easier, let's put everything in terms of $\sin$ and $\cos$ $\csc^2\theta = \dfrac{1}{\sin^2\theta}$ , so we can plug that in to get $ (\sin^2\theta)(\csc^2\theta) = \left(\sin^2\theta\right) \left(\dfrac{1}{\sin^2\theta}\right) $ This is $1$.